A014236 Expansion of g.f.: 2*x*(1-x)/((1-2*x)*(1-2*x^2)).

0, 2, 2, 8, 12, 32, 56, 128, 240, 512, 992, 2048, 4032, 8192, 16256, 32768, 65280, 131072, 261632, 524288, 1047552, 2097152, 4192256, 8388608, 16773120, 33554432, 67100672, 134217728, 268419072, 536870912, 1073709056, 2147483648, 4294901760, 8589934592

Offset: 0

Author

Paul F. Hudrlik (hudrlik(AT)scs.howard.edu)

Keywords

• nonn
• easy

Comments

Number of symmetric chiral (optically active) isomers possible for organic compounds with n distinct carbon atoms.

A020522 interleaved with A004171 and apparently the number of asymmetric Dyck (n+2)-paths with exactly half of the steps lying between the first and last peaks; e.g. all asymmetric 3-paths (UU*DDU*D and U*DUU*DD) comply so a(1)=2. - David Scambler, Sep 14 2012

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Crossrefs

Second differences of A027556.

Programs

GAP

a:=[0,2,2];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-4*a[n-3]; od; a; # G. C. Greubel, Jun 22 2019

Magma

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( 2*x*(1-x)/((1-2*x)*(1-2*x^2)) )); // G. C. Greubel, Jun 22 2019

Maple

f := n -> if n mod 2 = 0 then 2^n-2^(n/2) else 2^n; fi;

Mathematica

CoefficientList[Series[2x (1-x)/((1-2x)(1-2x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,-4},{0,2,2},30] (* Harvey P. Dale, Dec 04 2018 *)

PARI

my(x='x+O('x^30)); concat([0], Vec(2*x*(1-x)/((1-2*x)*(1-2*x^2)))) \\ G. C. Greubel, Jun 22 2019

Sage

(2*x*(1-x)/((1-2*x)*(1-2*x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 22 2019

Formula

a(n) = 2*A007179(n). - R. J. Mathar, Nov 14 2011

From G. C. Greubel, Jun 22 2019: (Start)

a(n) = 2^((n - 2)/2)*(2^((n + 2)/2) - 1 - (-1)^n).

E.g.f.: exp(2*x) - cosh(sqrt(2)*x). (End)

Extensions

G.f. corrected by Olivier Gérard, Nov 13 2011